2004 Fiscal Year Final Research Report Summary
A synthetic study of positive solutions to elliptic and parabolic partial differential equations
| Project/Area Number |
13440042
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | TOKYO INSTITUTE OF TECHNOLOGY |
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Principal Investigator |
MURATA Minoru Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (50087079)
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| Co-Investigator(Kenkyū-buntansha) |
SHIGA Hiroshige Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (10154189)
KAWANAGO Tadashi Tokyo Institute of Technology, Graduate School of Science and Engineering, Associate Professor, 大学院・理工学研究科, 助教授 (20214661)
AIKAWA Hiroaki Shimane University, Interdisciplinary Faculty of Science and Engineering, Professor, 総合理工学部, 教授 (20137889)
ISHIGE Kazuhiro Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90272020)
KURATA Kazuhiro Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10186489)
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| Project Period (FY) |
2001 – 2004
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| Keywords | parabolic equation / elliptic equation / positive solution / fundamental solution / Green function / uniqueness / Martin boundary / potential theory |
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| Research Abstract |
M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations on Riemannian manifolds and domains of R^n, and gave a sharp and general sufficient condition for the uniqueness via asymptotic properties at infinity of the equation and manifolds (Ann.Scuola Normale Sup. Pisa,30(2001),171-223). This is a simple and general result which unifies all previous results on the uniqueness. By introducing tne notion of heat escape, M.Murata also gave a sharp and general sufficient condition for the non-uniqueness (Math.Ann.,327(2003),203-226).
M.Murata investigated the structure of positive solutions of second order elliptic equations in skew product form, and determined the Martin boundary and Martin kernel by exploiting and developing perturbation theory, estimates of fundamental solutions for parabolic equations, and uniqueness theorems for nonnegative solutions of parabolic equations (J.Func.Anal.,194(2002),53-141;J.Math.Soc.Japan,57(2005),1-27).
M.Murata and T.tsuchida gave the asymptotics at infinity of the Green function for an elliptic equation with periodic coefficients on R^n, and explicitly determined the Martin boundary for it (J.Diff.Eq.,195(2003),82-118). This solves an open problem by S.Agmon since 1984.
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